A discrete algebraic framework for stochastic systems which yield unique and exact solutions.

Many physical systems exhibit random or stochastic components which shape or even drive their dynamic behavior. The stochastic models and equations describing such systems are typically assessed numerically, with a few exceptions allowing for a mathematically more rigorous treatment in the framework of stochastic calculus. However, even if exact solutions can be obtained in special cases, some results remain ambiguous due to the analytical foundation on which this calculus rests. In this work, we set out to identify the conceptual problem which renders stochastic calculus ambiguous, and exemplify a discrete algebraic framework which, for all practical intents and purposes, not just yields unique and exact solutions, but might also be capable of providing solutions to a much wider class of stochastic models.

Aspects of randomness in neural graph structures.

In the past two decades, significant advances have been made in understanding the structural and functional properties of biological networks, via graph-theoretic analysis. In general, most graph-theoretic studies are conducted in the presence of serious uncertainties, such as major undersampling of the experimental data. In the specific case of neural systems, however, a few moderately robust experimental reconstructions have been reported, and these have long served as fundamental prototypes for studying connectivity patterns in the nervous system. In this paper, we provide a comparative analysis of these "historical" graphs, both in their directed (original) and symmetrized (a common preprocessing step) forms, and provide a set of measures that can be consistently applied across graphs (directed or undirected, with or without self-loops). We focus on simple structural characterizations of network connectivity and find that in many measures, the networks studied are captured by simple random graph models. In a few key measures, however, we observe a marked departure from the random graph prediction. Our results suggest that the mechanism of graph formation in the networks studied is not well captured by existing abstract graph models in their first- and second-order connectivity.

Algebraic approach to small-world network models.

We introduce an analytic model for directed Watts-Strogatz small-world graphs and deduce an algebraic expression of its defining adjacency matrix. The latter is then used to calculate the small-world digraph's asymmetry index and clustering coefficient in an analytically exact fashion, valid nonasymptotically for all graph sizes. The proposed approach is general and can be applied to all algebraically well-defined graph-theoretical measures, thus allowing for an analytical investigation of finite-size small-world graphs.

Analytical integrate-and-fire neuron models with conductance-based dynamics and realistic postsynaptic potential time course for event-driven simulation strategies.

In a previous paper (Rudolph & Destexhe, 2006), we proposed various models, the gIF neuron models, of analytical integrate-and-fire (IF) neurons with conductance-based (COBA) dynamics for use in event-driven simulations. These models are based on an analytical approximation of the differential equation describing the IF neuron with exponential synaptic conductances and were successfully tested with respect to their response to random and oscillating inputs. Because they are analytical and mathematically simple, the gIF models are best suited for fast event-driven simulation strategies. However, the drawback of such models is they rely on a nonrealistic postsynaptic potential (PSP) time course, consisting of a discontinuous jump followed by a decay governed by the membrane time constant. Here, we address this limitation by conceiving an analytical approximation of the COBA IF neuron model with the full PSP time course. The subthreshold and suprathreshold response of this gIF4 model reproduces remarkably well the postsynaptic responses of the numerically solved passive membrane equation subject to conductance noise, while gaining at least two orders of magnitude in computational performance. Although the analytical structure of the gIF4 model is more complex than that of its predecessors due to the necessity of calculating future spike times, a simple and fast algorithmic implementation for use in large-scale neural network simulations is proposed.

High-resolution intracellular recordings using a real-time computational model of the electrode.

Intracellular recordings of neuronal membrane potential are a central tool in neurophysiology. In many situations, especially in vivo, the traditional limitation of such recordings is the high electrode resistance and capacitance, which may cause significant measurement errors during current injection. We introduce a computer-aided technique, Active Electrode Compensation (AEC), based on a digital model of the electrode interfaced in real time with the electrophysiological setup. The characteristics of this model are first estimated using white noise current injection. The electrode and membrane contribution are digitally separated, and the recording is then made by online subtraction of the electrode contribution. Tests performed in vitro and in vivo demonstrate that AEC enables high-frequency recordings in demanding conditions, such as injection of conductance noise in dynamic-clamp mode, not feasible with a single high-resistance electrode until now. AEC should be particularly useful to characterize fast neuronal phenomena intracellularly in vivo.

Characterizing synaptic conductance fluctuations in cortical neurons and their influence on spike generation.

Cortical neurons are subject to sustained and irregular synaptic activity which causes important fluctuations of the membrane potential (V(m)). We review here different methods to characterize this activity and its impact on spike generation. The simplified, fluctuating point-conductance model of synaptic activity provides the starting point of a variety of methods for the analysis of intracellular V(m) recordings. In this model, the synaptic excitatory and inhibitory conductances are described by Gaussian-distributed stochastic variables, or "colored conductance noise". The matching of experimentally recorded V(m) distributions to an invertible theoretical expression derived from the model allows the extraction of parameters characterizing the synaptic conductance distributions. This analysis can be complemented by the matching of experimental V(m) power spectral densities (PSDs) to a theoretical template, even though the unexpected scaling properties of experimental PSDs limit the precision of this latter approach. Building on this stochastic characterization of synaptic activity, we also propose methods to qualitatively and quantitatively evaluate spike-triggered averages of synaptic time-courses preceding spikes. This analysis points to an essential role for synaptic conductance variance in determining spike times. The presented methods are evaluated using controlled conductance injection in cortical neurons in vitro with the dynamic-clamp technique. We review their applications to the analysis of in vivo intracellular recordings in cat association cortex, which suggest a predominant role for inhibition in determining both sub- and supra-threshold dynamics of cortical neurons embedded in active networks.

Activated cortical states: experiments, analyses and models.

In awake animals, the cerebral cortex displays an "activated" state, with distinct characteristics compared to other states like slow-wave sleep or anesthesia. These characteristics include a sustained depolarized membrane potential (V(m)) and irregular firing activity. In the present paper, we evaluate our understanding of cortical activated states from a computational neuroscience point of view. We start by reviewing the electrophysiological characteristics of activated cortical states based on recordings and analysis performed in awake cat association cortex. These analyses show that cortical activity is characterized by an apparent Poisson-distributed stochastic dynamics, both at the single-cell and population levels, and that single cells display a high-conductance state dominated by inhibition. We next overview computational models of the "awake" cortex, and perform the same analyses as in the experiments. Many properties identified experimentally are indeed reproduced by models, such as depolarized V(m), irregular firing with apparent Poisson statistics, and the determinant role of inhibitory fluctuations on spiking. However, other features are not well reproduced, such as firing statistics and the conductance state of the membrane, suggesting that the network state displayed by models is not entirely correct. We also show how networks can approach a correct conductance state, suggesting ways by which future models will generate activity fully consistent with experimental data.